Kinematics of Rigid Bodies

Learning Objectives

  • Define and categorize different types of rigid body motion.
  • Apply angular kinematic equations for rotation about a fixed axis.
  • Determine velocity and acceleration of a point on a rotating body.
  • Analyze relative motion using translating and rotating axes.
  • Calculate velocities using the Instantaneous Center of Zero Velocity (IC).

While particle kinematics ignores rotational effects, rigid body kinematics accounts for both the translation and rotation of bodies. A rigid body is idealized as a system of particles wherein the distance between any two specific particles remains perfectly constant under applied loads.

Types of Rigid Body Motion

Rigid Body Motion Categories

  • Translation: Any straight line segment drawn on the body remains parallel to its original orientation throughout the motion. All particles in the body have the exact same velocity and acceleration.
    • Rectilinear Translation: Paths of particles are straight lines.
    • Curvilinear Translation: Paths of particles are congruent curves.
  • Rotation About a Fixed Axis: All particles move in circular paths centered on the axis of rotation, except for those lying on the axis (which have zero velocity).
  • General Plane Motion: A combination of translation and rotation. The body undergoes a displacement consisting of a translation within a reference plane and a rotation about an axis perpendicular to that plane.

Interactive Simulation

Interact with the simulation below to explore rigid body translation.

Rigid Body Translation & Rotation

Motion Control

Kinematics Equation Solver

Governing Principle:

In translation, all lines in the body remain parallel. Thus:

vA=vBandaA=aB\mathbf{v}_A = \mathbf{v}_B \quad \text{and} \quad \mathbf{a}_A = \mathbf{a}_B
Live Values:
Point A
v: 96.0 m/s
a: 0.0 m/s²
Point B
v: 96.0 m/s
a: 0.0 m/s²
✓ Verified: Point A and Point B have identical motion vectors.
Rigid Body Motion Trajectory WorkspaceABv_Av_B
Velocity at A (v_A)
Velocity at B (v_B)
Acceleration (a_A)
Acceleration (a_B)

Rotation About a Fixed Axis

When a rigid body rotates about a fixed axis, its motion is described by angular parameters.

Angular Position (θ\theta)

The angle defining the position of a line on the body, usually measured in radians.

Angular Velocity Equation

Defines angular velocity as the time rate of change of angular position.

ω=dθdt\omega = \frac{d\theta}{dt}

Variables

SymbolDescriptionUnit
ω\omegaAngular velocityrad/s
θ\thetaAngular positionrad
ttTimes

Angular Acceleration Equation

Defines angular acceleration as the time rate of change of angular velocity.

α=dωdt=d2θdt2\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}

Variables

SymbolDescriptionUnit
α\alphaAngular accelerationrad/s²
ω\omegaAngular velocityrad/s
θ\thetaAngular positionrad
ttTimes

Differential Angular Kinematics

Similar to linear motion, eliminating dtdt yields a purely spatial relation for angular kinematics.

Differential Angular Kinematic Equation

Relates angular acceleration, position, and velocity independent of time.

α dθ=ω dω\alpha \, d\theta = \omega \, d\omega

Variables

SymbolDescriptionUnit
α\alphaAngular accelerationrad/s²
dθd\thetaDifferential change in angular positionrad
ω\omegaAngular velocityrad/s
dωd\omegaDifferential change in angular velocityrad/s

If angular acceleration αc\alpha_c is constant, we obtain angular equations entirely analogous to linear constant acceleration formulas.

Constant Angular Acceleration Equations

Equations of motion for a rigid body rotating with a constant angular acceleration.

ω=ω0+αct\omega = \omega_0 + \alpha_c tθ=θ0+ω0t+12αct2\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha_c t^2ω2=ω02+2αc(θ−θ0)\omega^2 = \omega_0^2 + 2\alpha_c(\theta - \theta_0)

Variables

SymbolDescriptionUnit
ω\omegaFinal angular velocityrad/s
ω0\omega_0Initial angular velocityrad/s
αc\alpha_cConstant angular accelerationrad/s²
θ\thetaFinal angular positionrad
θ0\theta_0Initial angular positionrad
ttTimes

Velocity and Acceleration of a Point

Point Kinematics in Rotation

For a point PP located at a distance rr from the axis of rotation, its kinematics can be determined directly from the angular parameters.

Point Kinematics Equations

Relates linear velocity and acceleration components to angular parameters.

v=ωrv = \omega rat=αra_t = \alpha ran=ω2r=v2ra_n = \omega^2 r = \frac{v^2}{r}

Variables

SymbolDescriptionUnit
vvVelocity magnitude, tangent to the circular pathm/s
ω\omegaAngular velocityrad/s
rrDistance from the axis of rotationm
ata_tTangential acceleration magnitudem/s²
α\alphaAngular accelerationrad/s²
ana_nNormal acceleration magnitude, directed towards the axism/s²
Point Kinematics in Fixed-Axis Rotation

Connect angular motion to a point's tangent velocity and its two acceleration components.

The lesson's equations give the magnitudes v = ωr, a_t = αr, and a_n = ω²r; directions follow the right-hand rule and the inward normal direction.
Rotation about a fixed axis defines angular velocity and acceleration
Point velocity is tangent to its circular path
Acceleration combines tangential change and inward normal acceleration

Relative Motion Analysis Using Translating Axes

A powerful method for analyzing general plane motion is to break the motion down into two parts: a translation of a chosen base point, and a pure rotation about that base point. This relies on non-rotating (translating) axes attached to the base point.

Relative Velocity Concept

Let points A and B lie on the same rigid body undergoing general plane motion. A translating reference frame is attached to base point A. The velocity of B equals the velocity of A plus the velocity of B relative to A due to rotation.

Relative Velocity Equation

Calculates the velocity of point B using a translating reference frame at point A.

vB=vA+ω×rB/A\mathbf{v}_B = \mathbf{v}_A + \mathbf{\omega} \times \mathbf{r}_{B/A}

Variables

SymbolDescriptionUnit
vB\mathbf{v}_BVelocity of point Bm/s
vA\mathbf{v}_AVelocity of base point Am/s
ω\mathbf{\omega}Angular velocity vector of the rigid bodyrad/s
rB/A\mathbf{r}_{B/A}Position vector of B with respect to Am

Relative Acceleration Concept

The acceleration of B equals the acceleration of A plus the relative acceleration of B with respect to A. The relative acceleration consists of tangential and normal components.

Relative Acceleration Equation

Calculates the acceleration of point B using a translating reference frame at point A.

aB=aA+α×rB/A−ω2rB/A\mathbf{a}_B = \mathbf{a}_A + \mathbf{\alpha} \times \mathbf{r}_{B/A} - \omega^2 \mathbf{r}_{B/A}

Variables

SymbolDescriptionUnit
aB\mathbf{a}_BAcceleration of point Bm/s²
aA\mathbf{a}_AAcceleration of base point Am/s²
α\mathbf{\alpha}Angular acceleration vector of the rigid bodyrad/s²
rB/A\mathbf{r}_{B/A}Position vector of B with respect to Am
ω\omegaMagnitude of angular velocityrad/s
Relative Motion for Two Points on One Body

Separate base-point motion from the rotation that carries point A to point B.

The velocity equation adds ω × r_B/A. The acceleration equation adds α × r_B/A and the inward term −ω²r_B/A.
Start with the absolute motion of base point A
Add B's rotational motion relative to A
Use tangential and normal terms for B's relative acceleration

Relative Motion Analysis Using Rotating Axes

While translating axes are sufficient for simple problems where points A and B lie on the same rigid body, many mechanisms involve sliding contacts on rotating links. In these cases, it's necessary to express relative motion using a coordinate system (x,y)(x,y) that is attached to a rotating body.

Coriolis Acceleration Concept

When a particle moves with a velocity relative to a frame rotating with an angular velocity, an extra acceleration component arises, known as the Coriolis acceleration (ac\mathbf{a}_c).

Relative Velocity with Rotating Axes

Calculates velocity when the reference frame is itself rotating.

vB=vA+Ω×rB/A+(vB/A)xyz\mathbf{v}_B = \mathbf{v}_A + \mathbf{\Omega} \times \mathbf{r}_{B/A} + (\mathbf{v}_{B/A})_{xyz}

Variables

SymbolDescriptionUnit
vB\mathbf{v}_BAbsolute velocity of point Bm/s
vA\mathbf{v}_AAbsolute velocity of origin A of moving framem/s
Ω\mathbf{\Omega}Angular velocity of the rotating framerad/s
rB/A\mathbf{r}_{B/A}Position of B relative to Am
(vB/A)xyz(\mathbf{v}_{B/A})_{xyz}Velocity of B relative to the moving framem/s

Relative Acceleration with Rotating Axes (Coriolis)

Calculates acceleration when the reference frame is rotating, including the Coriolis component.

aB=aA+Ω˙×rB/A+Ω×(Ω×rB/A)+2Ω×(vB/A)xyz+(aB/A)xyz\mathbf{a}_B = \mathbf{a}_A + \dot{\mathbf{\Omega}} \times \mathbf{r}_{B/A} + \mathbf{\Omega} \times (\mathbf{\Omega} \times \mathbf{r}_{B/A}) + 2\mathbf{\Omega} \times (\mathbf{v}_{B/A})_{xyz} + (\mathbf{a}_{B/A})_{xyz}

Variables

SymbolDescriptionUnit
aB\mathbf{a}_BAbsolute acceleration of point Bm/s²
aA\mathbf{a}_AAbsolute acceleration of origin Am/s²
Ω˙\dot{\mathbf{\Omega}}Angular acceleration of the rotating framerad/s²
Ω\mathbf{\Omega}Angular velocity of the rotating framerad/s
rB/A\mathbf{r}_{B/A}Position of B relative to Am
(vB/A)xyz(\mathbf{v}_{B/A})_{xyz}Velocity of B relative to moving framem/s
(aB/A)xyz(\mathbf{a}_{B/A})_{xyz}Acceleration of B relative to moving framem/s²
Rotating-Frame Acceleration Contributions

Read the absolute acceleration as contributions from frame motion and motion relative to the frame.

The Coriolis contribution is 2Ω × v_rel. It is present in the acceleration relation when the point moves relative to the rotating frame.
1
Origin acceleration
2
Frame angular acceleration: Euler term
3
Frame rotation: centripetal term
4
Relative velocity in a rotating frame: Coriolis term
5
Relative acceleration in the frame

Coriolis Acceleration Direction

A common mistake is applying the wrong sign or direction to the Coriolis acceleration component: 2Ω×(vB/A)xyz2\mathbf{\Omega} \times (\mathbf{v}_{B/A})_{xyz}. Because it is a cross product, the direction is strictly determined by the right-hand rule. Rotate the vector Ω\mathbf{\Omega} into the vector (vB/A)xyz(\mathbf{v}_{B/A})_{xyz}; your thumb points in the direction of the Coriolis acceleration.

General Plane Motion: Instantaneous Center of Zero Velocity (IC)

General plane motion can be viewed at any specific instant as a pure rotation about a single, unique axis. The point where this axis intersects the plane of motion is called the Instantaneous Center of Zero Velocity (IC).

Limitations of the IC Method

The IC is a point in the plane of motion that momentarily has zero velocity (vIC=0v_{IC} = 0). By locating the IC, the velocity of any other point AA on the body can be found simply as vA=ω⋅rA/ICv_A = \omega \cdot r_{A/IC}, with the direction of vAv_A perpendicular to the line connecting AA and the IC.

Note: The IC only has zero velocity at that instant. Its acceleration is generally not zero. Therefore, the IC method is strictly for velocity analysis, not acceleration analysis.

Locating the Instantaneous Center of Zero Velocity (IC)

  1. Given Velocity Directions: If the directions of the velocities of two points AA and BB are known, and they are not parallel, draw lines perpendicular to vA\mathbf{v}_A and vB\mathbf{v}_B. The intersection of these lines is the IC.
  2. Given Parallel Velocities (Perpendicular to AB): If vA\mathbf{v}_A and vB\mathbf{v}_B are parallel and perpendicular to the line segment connecting AA and BB, draw a line connecting the heads of the velocity vectors and another connecting their tails (the line ABAB). The intersection of these two lines is the IC.
  3. Given Velocity and Angular Velocity: If the velocity vA\mathbf{v}_A of a point and the angular velocity ω\omega of the body are known, the IC lies on the line perpendicular to vA\mathbf{v}_A at a distance r=vA/ωr = v_A / \omega.
Rigid-Body Kinematics Method Selection

Select fixed-axis, instantaneous-center, or relative-motion equations from the requested motion quantities and constraints.

The instantaneous center is a velocity construction for one instant: it has zero velocity then, but its acceleration is generally nonzero. Do not use the IC method to determine acceleration. A point sliding relative to a rotating frame uses the rotating-frame relative-motion equations; include the Coriolis term when determining acceleration, not when using the relative-velocity equation.

Rigid-Body Kinematics Method SelectionSelect fixed-axis, instantaneous-center, or relative-motion equations from the requested motion quantities and constraints.. The instantaneous center is a velocity construction for one instant: it has zero velocity then, but its acceleration is generally nonzero. Do not use the IC method to determine acceleration. A point sliding relative to a rotating frame uses the rotating-frame relative-motion equations; include the Coriolis term when determining acceleration, not when using the relative-velocity equation.. Start kinematics analysis → Record constraints, known data, and requested quantities; Record constraints, known data, and requested quantities → Does the body rotate about a fixed axis?; Does the body rotate about a fixed axis? — Yes → Use fixed-axis angular and point-kinematics equations; Does the body rotate about a fixed axis? — No → Are only point velocities required?; Are only point velocities required? — Yes → Can the IC be located from available constraints?; Are only point velocities required? — No → Are the points fixed on the same rigid body?; Can the IC be located from available constraints? — Yes → Use the IC for velocities at this instant only; Can the IC be located from available constraints? — No → Use relative velocity for remaining point unknowns; Use relative velocity for remaining point unknowns → Are the points fixed on the same rigid body?; Are the points fixed on the same rigid body? — Yes → Use same-body translating-frame relations; Are the points fixed on the same rigid body? — No → Use rotating-frame relative-motion equations; Use fixed-axis angular and point-kinematics equations → Apply motion constraints; check vectors and units; Use the IC for velocities at this instant only → Apply motion constraints; check vectors and units; Use same-body translating-frame relations → Apply motion constraints; check vectors and units; Use rotating-frame relative-motion equations → Apply motion constraints; check vectors and units; Apply motion constraints; check vectors and units → Kinematic state determined

Start kinematics analysis → Record constraints, known data, and requested quantities; Record constraints, known data, and requested quantities → Does the body rotate about a fixed axis?; Does the body rotate about a fixed axis? — Yes → Use fixed-axis angular and point-kinematics equations; Does the body rotate about a fixed axis? — No → Are only point velocities required?; Are only point velocities required? — Yes → Can the IC be located from available constraints?; Are only point velocities required? — No → Are the points fixed on the same rigid body?; Can the IC be located from available constraints? — Yes → Use the IC for velocities at this instant only; Can the IC be located from available constraints? — No → Use relative velocity for remaining point unknowns; Use relative velocity for remaining point unknowns → Are the points fixed on the same rigid body?; Are the points fixed on the same rigid body? — Yes → Use same-body translating-frame relations; Are the points fixed on the same rigid body? — No → Use rotating-frame relative-motion equations; Use fixed-axis angular and point-kinematics equations → Apply motion constraints; check vectors and units; Use the IC for velocities at this instant only → Apply motion constraints; check vectors and units; Use same-body translating-frame relations → Apply motion constraints; check vectors and units; Use rotating-frame relative-motion equations → Apply motion constraints; check vectors and units; Apply motion constraints; check vectors and units → Kinematic state determined

  • Start kinematics analysis: terminator
  • Record constraints, known data, and requested quantities: process
  • Does the body rotate about a fixed axis?: decision
  • Use fixed-axis angular and point-kinematics equations: process
  • Are only point velocities required?: decision
  • Can the IC be located from available constraints?: decision
  • Use the IC for velocities at this instant only: process
  • Use relative velocity for remaining point unknowns: process
  • Are the points fixed on the same rigid body?: decision
  • Use same-body translating-frame relations: process
  • Use rotating-frame relative-motion equations: process. For acceleration include the Coriolis term; for velocity use the rotating-frame velocity relation.
  • Apply motion constraints; check vectors and units: process
  • Kinematic state determined: terminator

Interactive Simulation

Interact with the simulation below to explore the Instantaneous Center of Zero Velocity.

Instantaneous Center of Zero Velocity (IC) Visualizer

Mechanism Setup

s < 0 (Braking)s = 0 (Pure)s > 0 (Spin/Skid)

Mathematical Equations

Wheel IC Location & Slip Mechanics:

IC offset from center O is given by slip ratio:

yIC−yO=R(1+s)=2.0(1+0.00)=2.00 my_{IC} - y_O = R(1 + s) = 2.0(1 + 0.00) = 2.00\text{ m}

Pure rolling: IC is exactly at the point of contact.

Dragged Point P Velocity:
vP=ω×rP/ICv_P = \omega \times r_{P/IC}
d from IC: 3.27 m|v_P: 6.55 m/s
Instantaneous Center Geometric WorkspaceCenter OContactv_Ov_AICP (Drag)v_P
💡 Drag point P to analyze local velocity.
IC (Instant Center, v=0)
Velocity vector v_P
Distance ray r_P/IC
Key Takeaways
  • Rotation About a Fixed Axis: Governed by equations ω=dθ/dt\omega = d\theta/dt and α=dω/dt\alpha = d\omega/dt.
  • Point Acceleration: Has normal (an=ω2ra_n = \omega^2 r) and tangential (at=αra_t = \alpha r) components.
  • General Plane Motion is the superposition of translation of a reference point and rotation about that point.
  • Relative Velocity and Acceleration use vector equations (vB=vA+ω×r\mathbf{v}_B = \mathbf{v}_A + \mathbf{\omega} \times \mathbf{r}) to relate points on the same rigid body.
  • Instantaneous Center (IC): A powerful technique to find velocities by treating general plane motion instantaneously as pure rotation about the IC.